Eigenfunction expansions and spacetime estimates for generators in divergence-form

Let H = -∑ⁿ _j,k=1∂/∂x_j be a formally self-adjoint (elliptic) operator in L² (ℝⁿ), n < 2. The real coefficients a_{j, k}(x) = a_{k, j}(x) are assumed to be bounded and to coincide with -Δ outside of a ball. The paper deals with two topics: (i) An eigenfunction expansion theorem, proving in particular that H is unitarily equivalent to -Δ, and (ii) Global spacetime estimates for the associated inhomogeneous wave equation, proved under suitable ("nontrapping") additional assumptions on the coefficients. The main tool used here is a Limiting Absorption Principle (LAP) in the framework of weighted Sobolev spaces, which holds also at the threshold.